Deriving the symbol error rate

Where the additive noise follows the Gaussian probability distribution function,

with and .

The conditional probability distribution function (PDF) of received symbol given was transmitted is:

.

As can be seen from the figure above, due to the addition of noise, the transmitted symbol gets spreaded. However, if the received symbol is present with in the boundary defined by the magenta lines, then the symbol will be demodulated correctly.

To derive the symbol error rate, the objective is to find the probability that the phase of the received symbol lies within this boundary defined by the magenta lines i.e. from to .

For simplifying the derivation, let us make the following assumptions:

(a) The signal to noise ratio, is reasonably high.

For a reasonably high value of , then the real part of the received symbol is not afected by noise i.e.,

and

the imaginary part of the received symbol is equal to noise, i.e..

(b) The value of M is reasonably high (typically M >4 suffice)

For a reasonably high value of M, the constellation points are closely spaced. Given so, the distance of the constellation point to the magenta line can be approximated as .

Figure: Distance between constellation points

Given the above two assumptions, it can be observed that the symbol will be decoded incorrectly, if the imaginary component of received symbol isgreater than . The probability of being greater than is,

.

Changing the variable to ,

.

Note: The complementary error function, .

Similarly, the symbol will be decoded incorrectly, if the imaginary component of received symbol is less than . The probability of being less than is,

.

The total probability of error given was transmittd is,

.

Total symbol error rate

The symbol will be in error, if atleast one of the symbol gets decoded incorrectly. Hence the total symbol error rate from M-PSK modulation is,

.

Simulation model

Simple Matlab/Octave script for simulating transmission and recepetion of an M-PSK modulation is attached. It can be observed that the simulated symbol error rate compares well with the theoretical symbol error rate.

Hi Sir,
In the code for converting Es_N0 from dB to linear you used “10^(-Es_N0_dB(ii)/20)”. But at the end for finding theoritical SER you used “theoryBer = erfc(sqrt(10.^(Es_N0_dB/10))*sin(pi/M));”. Please observe Es_N0_dB/10 here.
Can you explain why dB to linear conversion is different ways ?
Regards,
K. Sri Ram.

Hi Sir,
In simulations of BPSK – BER and QPSK -BER also you did the same thing of converting dB to linear in two ways.( while adding noise to signal used Es_N0/20 but at the end theoretical BER you used Es_N0/10). Please explain why ?

Hello, I have a problem that I need to solve about a
modulation algorithm. I’m working on a assignment and need a code in Matlab to the
in-phase and quadrature components of a 16 QAM. Afterwards, I will run the
program in a Vector Signal Analyzer and I will have to see the I and Q waveforms
and the constellation.

I have been trying out by creating an array with 16 values (0 to 15) and
converting them to binary (0000 to 1111), afterwards, I have created the I and Q
signals, as a sine or cosine of the array values mentioned respectively using a
for loop, thus creating the different values for the phase, but I don’t know how
to express the amplitude levels (-3, -1, 1, 3).

Hi Krishna,
i`m an MSc communication systems engineering student in the university of portsmouth, i would love to be a member of this forum and there is this problem i want to solve:
Analysis & Simulation of 16PSK Using
MATLAB.

OBJECTIVES:
· Demonstrate the ability to learn MATLAB for the purpose of this
coursework
· Write a technical report describing the principle, theory and applications
of 16PSK. Your system will be tested over the AWGN Channel.
· Implement a MATLAB simulation of 16PSK modulator and
demodulator system.
· Measurement of the symbol error rate and bit error rate as a function of
the ratios Eb / No and Es / No achieved by the system in the presence of
additive white Gaussian Noise. Your measurements should be compared
to the theoretical estimates that you should have derived from first
principles.
· Show that you are able to identify and use technical references.

But for now am learning MATLAB and the problem is giving me though time, i wish you can help me by putting me through with the rest apart from the MATLAB. Please sir i need your help.
Regards.
Tukuben

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@Rajesh:
1. If the device is non-linear, one would expect that the signal at the output will have more frequencies than the signal at the input (generation of harmonics dye to x^2, x^3 terms etc).

The out-of-band frequencies maybe filtered out, however nothing much can be done about the harmonics in the in-band. It may be that the measured power includes both out-band and in-band harmonics (along with the desired signal power), hence the C/N does not degrade. However, since the in-band harmonics are not deisrable for demodulation, there is degradation in the MER.

2. When you say ‘devices not matched’, I would think that you are referring to impedence mismatch between the devices (resulting in reflection…). May I suggest the generation of in-band and out-band frequencies due to impedence mismatch as a probable reason for MER degradation.

Thanks for the reply. While you are right, however we will have to add the noise level. Now the noise level at room temp will be about 3 dBuV while that for QPSK with MER of 15 dB at Signal strength of 86dBuV will be approx 71 dBuV. So that the additive White noise is negligible. We also find that C/N does not get deterioted as we pass thru varios devices.

However we have noticed that MER doe reduce by 0.7-1.0dB.

Question is

1. If thye device contains diodes then if at the point of operation diode is non linear it can add to the reduction of MER but not of C/N. How MER is related to non-linear CSO and CRB.
2. If the device is poorly matched then agian MER measurement will get affected. How does the mismatch between the device and the instruments affects the MER?

2. Is your question the following: If we pass a received symbol with a noise of xdB
through a splitter having resistive loss of 1dB, how much will be the resultant noise
power. Did I interpret correctly?
I would say the resultant noise will be 10*log10(10^(-x/10) + 10^(-1/10))
Did the linear addition of two noise components, do you agree?

1. The symbol error rate should also be related to Modulation Error Rate (MER). Since MER is what most Meters measure a correlation will be most helpful.

2. Most of the signal are distributed over a network. What one also needs is the MER impairments due to network elements.

For example if the signal is split into two parts. That splitter will have an insertion loss which will have a spatial component. For example a 2 way lossless splitter will reduce the signal and noise by 3 dB and hence will not impair the signal. However there is an associated resistive loss aboput 1 dB. How much will the MER deteoriate?

@mahesh:
Yeah… i agree. it takes a bit of time to get used to 20*log10() or 10*log10() …
My rule of thumb is as follows:
(a) for voltage signals use 20*log10()
(b) for power signals use 10*log10().

In the code, as you observed,
(a) for scaling noise, which is a voltage signal, I used the 20*log10().
(b) for finding the theoretical symbol error rate, the Es/No in dB (which is signal power by noise power), the conversion used is 10*log10().

In your simulation, you have added noise voltage/level i.e taken 20log10(ns) whereas for calculating theoretical BER , you have used noise power. (10log10(ns^2)). I always tend to get confused between these. Do you know any study material clarifying these and some examples?
thanks
mahesh

@mahesh: true. Infact in the transmit and receive chain, there are various sources which can introduce DC (LO leakage etc). Given so, it makes it difficult to recover information present on DC subcarrier.

in wimax ieee802.16e standard
for synchronization purpose they are using preamble .in that preamble they using dc subcarrier .why they are using dc subcarrier… please if any one knows clarify my dought…its urgent